Higher Dimensions
More general Heisenberg groups Hn may be defined for higher dimensions in Euclidean space, and more generally on symplectic vector spaces. The simplest general case is the real Heisenberg group of dimension 2n+1, for any integer n ≥ 1. As a group of matrices, Hn (or Hn(R) to indicate this is the Heisenberg group over the ring R or real numbers) is defined as the group of square matrices of size n+2 with entries in R:
where
- a is a row vector of length n,
- b is a column vector of length n,
- is the identity matrix of size n.
Read more about this topic: Heisenberg Group
Famous quotes containing the words higher and/or dimensions:
“They say the seeds of what we will do are in all of us, but it always seemed to me that in those who make jokes in life the seeds are covered with better soil and with a higher grade of manure.”
—Ernest Hemingway (1899–1961)
“I was surprised by Joe’s asking me how far it was to the Moosehorn. He was pretty well acquainted with this stream, but he had noticed that I was curious about distances, and had several maps. He and Indians generally, with whom I have talked, are not able to describe dimensions or distances in our measures with any accuracy. He could tell, perhaps, at what time we should arrive, but not how far it was.”
—Henry David Thoreau (1817–1862)